

where \( a_{p,m} \)  is maximum acceleration, \(\mathbf {\Phi }^{[\kappa -1]}(t) = [ \mathbf {\Phi }(t)^T, ... , \mathbf {\Phi }^{(\kappa -1)}(t)^T] \)  and \(\bar{\mathbf {\Phi }}_0, \bar{\mathbf {\Phi }}_f \in \mathbb {R}^{3\kappa } \)  are the initial and final conditions up to order \(\kappa = (m + 1)/2\) .
We employ the method in [1]} to represent the collision-free space \( \mathcal {F} \)  as a safe flight corridor and leverage the optimization framework in [2]} to relax the constraints and solve an unconstrained spatial-temporal trajectory optimization.
